Problem 71 - Precalculus: Functions and Graphs

Question

Express as a sum of terms of the form \(a x^{\prime}\) where \(r\) is a rational number. $$\frac{\left(x^{2}+2\right)^{2}}{x^{5}}$$

Step-by-Step Solution

Answer

\(x^{-1} + 4x^{-3} + 4x^{-5}\)

Step 1: Expand the Numerator

First, we need to expand the expression in the numerator \((x^2 + 2)^2\). To do this, use the formula \((a + b)^2 = a^2 + 2ab + b^2\). Here, \(a = x^2\) and \(b = 2\). Thus, the expanded form is \(x^4 + 4x^2 + 4\).

Step 2: Divide Each Term by the Denominator

Now, divide each term of the expanded numerator \(x^4 + 4x^2 + 4\) by the denominator \(x^5\). This gives us three fractions: \(\frac{x^4}{x^5}\), \(\frac{4x^2}{x^5}\), and \(\frac{4}{x^5}\).

Step 3: Simplify Each Fraction

Simplify each of the fractions independently: 1. \(\frac{x^4}{x^5} = x^{4-5} = x^{-1}\).2. \(\frac{4x^2}{x^5} = 4x^{2-5} = 4x^{-3}\).3. \(\frac{4}{x^5} = 4x^{-5}\).

Step 4: Combine the Simplified Terms

Combine the simplified terms to express the original expression in the form of a sum: \(x^{-1} + 4x^{-3} + 4x^{-5}\). This is the required expression where each term is of the form \(ax^r\).

Key Concepts

Exponent Rules

Understanding exponent rules is crucial when handling polynomial division and rational expressions. Exponents tell us how many times a number, called the base, is multiplied by itself. For example, \(x^2\) means \(x\) is multiplied by itself: \(x \times x\). Two essential exponent rules help simplify expressions: the product rule and the quotient rule.

• The **product rule** states that when multiplying like bases, you add their exponents: \(x^m \cdot x^n = x^{m+n}\).
• The **quotient rule** is used when dividing like bases. It tells us to subtract the exponent in the denominator from the numerator: \(\frac{x^m}{x^n} = x^{m-n}\).

These rules allowed us to transform each term of the original expression \(\frac{x^4}{x^5}\), \(\frac{4x^2}{x^5}\), and \(\frac{4}{x^5}\) into simpler forms like \(x^{-1}\), \(4x^{-3}\), and \(4x^{-5}\). Understanding these rules keeps equations simple and helps in solving polynomial division problems efficiently.

Rational Expressions

A rational expression is essentially a fraction where the numerator and denominator are polynomials. In our exercise, the expression \(\frac{(x^2 + 2)^2}{x^5}\) is a rational expression, with a polynomial on top and a power of \(x\) on the bottom.

Rational expressions can be tricky, especially when the numerators require expansion. Always ensure every possible polynomial in the numerator is fully expanded. In the given problem, first expanding \((x^2 + 2)^2\) into \(x^4 + 4x^2 + 4\) was crucial for simplifying each term separately.

When you work through rational expressions like these:

• Expand or factor the entire numerator, if necessary, before attempting to divide.
• Simplify each term independently by applying the exponent rules.
• Always check for common factors to further reduce if possible.

This way, you can effectively break complex rational expressions down into more manageable terms.

Polynomial Expansion

Polynomial expansion involves transforming an expression raised to a power into a sum of simpler terms. This technique is particularly useful for simplifying complex expressions before further manipulation.

The **binomial theorem** is a fundamental tool for expanding polynomial expressions like \((a + b)^n\). In our problem, \((x^2 + 2)^2\) was expanded using the formula \((a + b)^2 = a^2 + 2ab + b^2\), leading to \(x^4 + 4x^2 + 4\).

When expanding polynomials, consider:

• Identifying the terms \(a\) and \(b\) in the binomial.
• Applying the formula or theorem accurately to ensure each term is correctly developed.
• Once expanded, simplifying the resulting polynomial terms further as needed, like dividing each by \(x^5\) in this problem.

Using systematic expansion not only simplifies each step following but also aids in visualizing the distribution of coefficients and interaction between the terms. Thus, polynomial expansion is an essential step in dissecting and simplifying more advanced algebraic expressions.